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3 years ago

Use the quadratic formula to solve the equation 4x^2+6x-4=0

Mathematics

1 answer:

Vesnalui [34]3 years ago

4 0

Answer:

$\large\boxed{x=-2\ or\ x=\dfrac{1}{2}}$

Step-by-step explanation:

$\text{The quadratic formula of an equation}\ ax^2+bx+c=0\\\\x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\text{We have}\ 4x^2+6x-4=0\to a=4,\ b=6,\ c=-4.\\\\\text{Substitute:}\\\\x=\dfrac{-6\pm\sqrt{6^2-4(4)(-4)}}{2(4)}=\dfrac{-6\pm\sqrt{36+64}}{8}=\dfrac{-6\pm\sqrt{100}}{8}\\\\x=\dfrac{-6-10}{8}=\dfrac{-16}{8}=-2\ or\ x=\dfrac{-6+10}{8}=\dfrac{4}{8}=\dfrac{1}{2}$

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Please help me with the below question.

Snezhnost [94]

6a. By the convolution theorem,

$L\{t^3\star e^{5t}\} = L\{t^3\} \times L\{e^{5t}\} = \dfrac6{s^4} \times \dfrac1{s-5} = \boxed{\dfrac5{s^4(s-5)}}$

6b. Similarly,

$L\{e^{3t}\star \cos(t)\} = L\{e^{3t}\} \times L\{\cos(t)\} = \dfrac1{s-3} \times \dfrac s{1+s^2} = \boxed{\dfrac s{(s-3)(s^2+1)}}$

7. Take the Laplace transform of both sides, noting that the integral is the convolution of $e^t$ and $f(t)$ .

$\displaystyle f(t) = 3 - 4 \int_0^t e^\tau f(t - \tau) \, d\tau$

$\implies \displaystyle F(s) = \dfrac3s - 4 F(s) G(s)$

where $g(t) = e^t$ . Then $G(s) = \frac1{s-1}$ , and

$F(s) = \dfrac3s - \dfrac4{s-1} F(s) \implies F(s) = \dfrac{\frac3s}{\frac{s+3}{s-1}} = 3\dfrac{s-1}{s(s+3)}$

We have the partial fraction decomposition,

$\dfrac{s-1}{s(s+3)} = \dfrac13 \left(-\dfrac1s + \dfrac4{s+3}\right)$

Then we can easily compute the inverse transform to solve for f(t) :

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6 0

2 years ago

Translate the sentence into an inequality. The product of w and 3 is greater than 15.

Deffense [45]

Answer:

The product of w and 3 is greater than 15.

(w x 3)> 15

3w>15

Step-by-step explanation:

"Product" signifies multiplication. "of" means w is being multiplied by 3.

3 0

3 years ago

Help I will be marking brainliest!!!

Keith_Richards [23]

<h3>Answer: A. 18*sqrt(3)</h3>

=============================================

Explanation:

We'll need the tangent rule

tan(angle) = opposite/adjacent

tan(R) = TH/HR

tan(30) = TH/54

sqrt(3)/3 = TH/54 ... use the unit circle

54*sqrt(3)/3 = TH .... multiply both sides by 54

(54/3)*sqrt(3) = TH

18*sqrt(3) = TH

TH = 18*sqrt(3) which points to choice A as the final answer

----------------------

An alternative method:

Triangle THR is a 30-60-90 triangle.

Let x be the measure of side TH. This side is opposite the smallest angle R = 30, so we consider this the short leg.

The hypotenuse is twice as long as x, so TR = 2x. This only applies to 30-60-90 triangles.

Now use the pythagorean theorem

a^2 + b^2 = c^2

(TH)^2 + (HR)^2 = (TR)^2

(x)^2 + (54)^2 = (2x)^2

x^2 + 2916 = 4x^2

2916 = 4x^2 - x^2

3x^2 = 2916

x^2 = 2916/3

x^2 = 972

x = sqrt(972)

x = sqrt(324*3)

x = sqrt(324)*sqrt(3)

x = 18*sqrt(3) which is the length of TH.

A slightly similar idea is to use the fact that if y is the long leg and x is the short leg, then y = x*sqrt(3). Plug in y = 54 and isolate x and you should get x = 18*sqrt(3). Again, this trick only works for 30-60-90 triangles.

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3 years ago

Which expression is equivalent to 2^4 ⋅ 2^−7?

yaroslaw [1]